#300449

order	512
nilpotency class	3
isoclinism	isoclinic class of #300437
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2,7,3)(4,5,8,6)(9,10,16,12)(11,19,23,20)(13,15,18,14)(17,22,24,21);(1,3,7,2)(4,5,8,6)(9,11,15,22,16,23,14,21)(10,17,13,20,12,24,18,19);(1,4,5,3,7,8,6,2)(9,12,14,13,16,10,15,18)(11,19,22,17,23,20,21,24);(9,13,14,10,16,18,15,12)(11,20,21,17,23,19,22,24)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=f,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207645);